Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease.

Bivariate clustered (correlated) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed model (LMM) framework with underlying normality assumptions of the random effects and within-subject errors. However, such normality assumptions might be questionable if the data set particularly exhibits skewness and heavy tails. Using a Bayesian paradigm, we use the skew-normal/independent (SNI) distribution as a tool for modeling clustered data with bivariate non-normal responses in an LMM framework. The SNI distribution is an attractive class of asymmetric thick-tailed parametric structure which includes the skew-normal distribution as a special case. We assume that the random effects follow multivariate SNI distributions and the random errors follow SNI distributions which provides substantial robustness over the symmetric normal process in an LMM framework. Specific distributions obtained as special cases, viz. the skew-t, the skew-slash and the skew-contaminated normal distributions are compared, along with the default skew-normal density. The methodology is illustrated through an application to a real data which records the periodontal health status of an interesting population using periodontal pocket depth (PPD) and clinical attachment level (CAL).

[1]  Ernesto San Martín,et al.  Linear mixed models with skew-elliptical distributions: A Bayesian approach , 2008, Comput. Stat. Data Anal..

[2]  W. Borgnakke,et al.  Periodontal disease: associations with diabetes, glycemic control and complications. , 2008, Oral diseases.

[3]  R. Kass,et al.  Reference Bayesian Methods for Generalized Linear Mixed Models , 2000 .

[4]  R. Arellano-Valle,et al.  Bayesian Inference for Skew-normal Linear Mixed Models , 2007 .

[5]  Peter Müller,et al.  A Bayesian Population Model with Hierarchical Mixture Priors Applied to Blood Count Data , 1997 .

[6]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[7]  M. Walsh,et al.  Dental hygiene : theory and practice , 1994 .

[8]  Marc G. Genton,et al.  Skew-elliptical distributions and their applications : a journey beyond normality , 2004 .

[9]  M. Genton,et al.  On fundamental skew distributions , 2005 .

[10]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[11]  K. Lange,et al.  Normal/Independent Distributions and Their Applications in Robust Regression , 1993 .

[12]  S. Sahu,et al.  A new class of multivariate skew distributions with applications to bayesian regression models , 2003 .

[13]  R. Arellano-Valle,et al.  On the Unification of Families of Skew‐normal Distributions , 2006 .

[14]  A. Azzalini,et al.  Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution , 2003, 0911.2342.

[15]  Arjun K. Gupta,et al.  A multivariate skew normal distribution , 2004 .

[16]  Brian Burt,et al.  Position paper: epidemiology of periodontal diseases. , 2005, Journal of periodontology.

[17]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[18]  P. Reichard,et al.  The effect of long-term intensified insulin treatment on the development of microvascular complications of diabetes mellitus. , 1993, The New England journal of medicine.

[19]  Daniel Gianola,et al.  Robust Linear Mixed Models with Normal/Independent Distributions and Bayesian MCMC Implementation , 2003 .

[20]  M Davidian,et al.  Linear Mixed Models with Flexible Distributions of Random Effects for Longitudinal Data , 2001, Biometrics.

[21]  Victor H. Lachos,et al.  A robust Bayesian approach to null intercept measurement error model with application to dental data , 2009, Comput. Stat. Data Anal..

[22]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[23]  Rolando De la Cruz.  Bayesian non-linear regression models with skew-elliptical errors: Applications to the classification of longitudinal profiles , 2008, Comput. Stat. Data Anal..

[24]  B. Carlin,et al.  Spatial Analyses of Periodontal Data Using Conditionally Autoregressive Priors Having Two Classes of Neighbor Relations , 2007 .

[25]  G. Casella,et al.  The Effect of Improper Priors on Gibbs Sampling in Hierarchical Linear Mixed Models , 1996 .

[26]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[27]  E. Lesaffre,et al.  Smooth Random Effects Distribution in a Linear Mixed Model , 2004, Biometrics.

[28]  Tsung I. Lin,et al.  Robust mixture modeling using multivariate skew t distributions , 2010, Stat. Comput..

[29]  R. Arellano-Valle,et al.  LIKELIHOOD BASED INFERENCE FOR SKEW-NORMAL INDEPENDENT LINEAR MIXED MODELS , 2010 .

[30]  H Löe,et al.  Periodontal diseases in the United States population. , 1998, Journal of periodontology.

[31]  M. Genton,et al.  A unified view on skewed distributions arising from selections , 2006 .

[32]  G. Verbeke,et al.  A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population , 1996 .

[33]  Y. Matsuyama,et al.  Mixed models for bivariate response repeated measures data using Gibbs sampling. , 1997, Statistics in medicine.

[34]  A. Gelman.  Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper) , 2004 .

[35]  Health of Gullah Families in South Carolina With Type 2 Diabetes , 2009, The Diabetes educator.

[36]  Margaret Walsh Rdh Ms Ma EdD,et al.  Dental Hygiene Theory and Practice , 1995 .

[37]  A. Azzalini.  The Skew‐normal Distribution and Related Multivariate Families * , 2005 .

[38]  M. Wand,et al.  General design Bayesian generalized linear mixed models , 2006, math/0606491.

[39]  Alan E. Gelfand,et al.  Model choice: A minimum posterior predictive loss approach , 1998, AISTATS.